Let $R = L_1 \oplus L_2 \oplus ··· \oplus L_n$ where each $L_i$ is a left ideal and $n \in \mathbb{N}$. Show that there exist idempotents $e_i \in R$ where $e_i = e_i^2$ such that $1_R = \sum\limits_{i=1}^{n} e_i$, $L_i = Re_i$ and $e_i e_j = \delta_{i,j}e_i$ for all i, j.
All I really know that is the elements $\{e_i\}$ are called orthogonal idempotents. I am looking for hints on how to proceed not a premade solution.
How would I also start to prove a variant of this where $e_i \in Z(R)$?
Start by decomposing $1_R$ as a sum $\sum_{i=1}^ne_i$, where $e_i\in L_i$. This decomposition is unique, so you can actually be sure that these $e_i$'s must be the ones you want. Then, proceed to showing that they satisfy the conditions. You may deduce relations between the $e_i$'s by using the fact that $1_R$ is idempotent. Can you conclude with this?